The Field of a Line Charge Near the Tip of a Dielectric Wedge

Lewis and Wasserstrom 1 have calculated the strength of the field singularity at the tip of a dielectric wedge in the configuration shown in Fig. 1. In particular, with a wedge permittivity e x greater than the permittivity t> of the surrounding medium and a conductor angle /3 = 7r (the "overhanging electrode"), they found that the tip field was singular for all wedge angles a greater than t/2. From this analysis, it was concluded that semiconductor devices with undercut edges (a 7r/2) would be advantageous in reducing local field strength and thus preventing breakdown. Because the analysis of Ref. 1 was strictly local, based on an expansion of the potential in positive powers of the distance from the wedge vertex, multiplied by trigonometric functions of the polar angle, it was felt by some that the results were suspect, since they were not based on the solution of a complete boundary value problem. Here we lay that suspicion to rest by presenting the solution of such a problem, namely the field due to a line charge near a dielectric wedge, as shown in Fig. 2. The solution of this problem, previously treated by Smythe 2 in a somewhat involved fashion, gives Green's function for the composite region. Here we use the Mellin transform, obtaining an expansion of the potential near the wedge tip in terms of the poles of the transform. Based on this analysis, we conclude for the chargewedge configuration of Fig. 2 that, for arbitrary ratios e-i/ti, the wedge tip field is singular for all values of the half-angle a.